Ivan Mihajlin

Nondeterministic Extensions of the Strong Exponential Time Hypothesis and Consequences for Non-reducibility

We introduce the Nondeterministic Strong Exponential Time Hypothesis (NSETH) as a natural extension of the Strong Exponential Time Hypothesis (SETH). We show that both refuting and proving NSETH would have interesting consequences. In particular we show that disproving NSETH would give new nontrivial circuit lower bounds. On the other hand, NSETH implies non-reducibility results, i.e. the absence of (deterministic) fine-grained reductions from SAT to a number of problems. As a consequence we conclude that unless this hypothesis fails, problems such as 3-SUM, APSP and model checking of a large class of first-order graph properties cannot be shown to be SETH-hard using deterministic or zero-error probabilistic reductions.

Approximating Shortest Superstring Problem Using de Bruijn Graphs

The best known approximation ratio for the shortest superstring problem is \(2\frac{11}{23}\) (Mucha, 2012). In this note, we improve this bound for the case when the length of all input strings is equal to r, for r ≤ 7. E.g., for strings of length 3 we get a \(1\frac{1}{3}\)-approximation. An advantage of the algorithm is that it is extremely simple both to implement and to analyze. Another advantage is that it is based on de Bruijn graphs. Such graphs are widely used in genome assembly (one of the most important practical applications of the shortest common superstring problem). At the same time these graphs have only a few applications in theoretical investigations of the shortest superstring problem.

Solving SCS for bounded length strings in fewer than 2 n steps.

Abstract It is still not known whether a shortest common superstring (SCS) of n input strings can be found faster than in O ⁎ ( 2 n ) time ( O ⁎ ( ⋅ ) suppresses polynomial factors of the input length). In this short note, we show that for any constant r, SCS for strings of length at most r can be solved in time O ⁎ ( 2 ( 1 − c ( r ) ) n ) where c ( r ) = ( 1 + 2 r 2 ) − 1 . For this, we introduce so-called hierarchical graphs that allow us to reduce SCS on strings of length at most r to the directed rural postman problem on a graph with at most k = ( 1 − c ( r ) ) n weakly connected components. One can then use a recent O ⁎ ( 2 k ) time algorithm by Gutin, Wahlstrom, and Yeo.

Solving 3-Superstring in 3 n /3 Time

In the shortest common superstring problem (SCS) one is given a set s 1, …, s n of n strings and the goal is to find a shortest string containing each s i as a substring. While many approximation algorithms for this problem have been developed, it is still not known whether it can be solved exactly in fewer than 2 n steps. In this paper we present an algorithm that solves the special case when all of the input strings have length 3 in time 3 n/3 and polynomial space. The algorithm generates a combination of a de Bruijn graph and an overlap graph, such that a SCS is then a shortest directed rural postman path (DRPP) on this graph. We show that there exists at least one optimal DRPP satisfying some natural properties. The algorithm works basically by exhaustive search, but on the reduced search space of such paths of size 3 n/3.

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given n-vertex graph G can be mapped to the vertices of a given h-vertex graph H such that each edge of G is mapped to an edge of H. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 2-CSP problem. In this paper, we prove several lower bounds for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound \(2^{\Omega \left( \frac{n \log h}{\log \log h}\right) }\). This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound \(2^{\mathcal {O}(n\log {h})}\) is almost asymptotically tight.

New Lower Bounds on Circuit Size of Multi-output Functions

AbstractLet Bn, m be the set of all Boolean functions from {0, 1}n to {0, 1}m, Bn = Bn, 1 and U2 = B2∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U2. 1.A lower bound CU2(f)≥5n−o(n)

Families with Infants: A General Approach to Solve Hard Partition Problems

C_{U_{2}}(f) \ge 5n-o(n)

Tight Lower Bounds on Graph Embedding Problems

for a linear function f ∈ Bn − 1,logn. The lower bound follows from the following more general result: for any matrix A ∈ {0, 1}m × n with n pairwise different non-zero columns and b ∈ {0, 1}m, CU2(Ax⊕b)≥5(n−m).

Hardness amplification for non-commutative arithmetic circuits

C_{U_{2}}(Ax \oplus b)\ge 5(n-m).